Combinatorics of transformations from standard to non-standard bases in Brauer algebras

Transformation coefficients between standard bases for irreducible representations of the Brauer centralizer algebra ${\mathfrak{B}}_f(x)$ and split bases adapted to the ${\mathfrak{B}}_{f_1}(x)\times {\mathfrak{B}}_{f_2}(x)\subset {\mathfrak{B}}_f(x)$ subalgebra ($f_1+f_2=f$) are considered. After providing the suitable combinatorial background, based on the definition of $i$-coupling relation on nodes of the subduction grid, we introduce a generalized version of the subduction graph which extends the one given in J. Phys. A: Math. Gen. $\mathbf{39}$ 7657-7668 for symmetric groups. Thus, we can describe the structure of the subduction system arising from the linear method and give an outline of the form of the solution space. An ordering relation on the grid is also given and then, as in the case of symmetric groups, the choices of the phases and of the free factors governing the multiplicity separations are discussed.